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Properties

Label 8619.2.a.bs
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $1$
Dimension $24$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.8230615021\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 11 q^{2} + 24 q^{3} + 29 q^{4} - 23 q^{5} - 11 q^{6} - 10 q^{7} - 27 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 11 q^{2} + 24 q^{3} + 29 q^{4} - 23 q^{5} - 11 q^{6} - 10 q^{7} - 27 q^{8} + 24 q^{9} + 8 q^{10} - 19 q^{11} + 29 q^{12} + 4 q^{14} - 23 q^{15} + 43 q^{16} + 24 q^{17} - 11 q^{18} - 20 q^{19} - 48 q^{20} - 10 q^{21} - 12 q^{22} - q^{23} - 27 q^{24} + 31 q^{25} + 24 q^{27} - 16 q^{28} + 11 q^{29} + 8 q^{30} - 42 q^{31} - 60 q^{32} - 19 q^{33} - 11 q^{34} - 25 q^{35} + 29 q^{36} - 20 q^{37} - 13 q^{38} + 55 q^{40} - 35 q^{41} + 4 q^{42} + 23 q^{43} - 29 q^{44} - 23 q^{45} - 4 q^{46} - 67 q^{47} + 43 q^{48} + 2 q^{49} - 41 q^{50} + 24 q^{51} - 7 q^{53} - 11 q^{54} + 22 q^{55} - 4 q^{56} - 20 q^{57} - 7 q^{58} - 45 q^{59} - 48 q^{60} - 10 q^{61} - 20 q^{62} - 10 q^{63} + 25 q^{64} - 12 q^{66} - 12 q^{67} + 29 q^{68} - q^{69} + 26 q^{70} - 51 q^{71} - 27 q^{72} - 18 q^{73} - 4 q^{74} + 31 q^{75} - 34 q^{76} - 9 q^{77} + 14 q^{79} - 74 q^{80} + 24 q^{81} + 23 q^{82} - 48 q^{83} - 16 q^{84} - 23 q^{85} - 68 q^{86} + 11 q^{87} - 38 q^{88} - 66 q^{89} + 8 q^{90} - 18 q^{92} - 42 q^{93} + 71 q^{94} + 10 q^{95} - 60 q^{96} - 15 q^{98} - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.81877 1.00000 5.94549 −3.59879 −2.81877 3.49305 −11.1214 1.00000 10.1442
1.2 −2.59398 1.00000 4.72871 −2.31950 −2.59398 −0.128462 −7.07822 1.00000 6.01672
1.3 −2.54973 1.00000 4.50115 1.81189 −2.54973 −2.54115 −6.37726 1.00000 −4.61984
1.4 −2.52567 1.00000 4.37901 −1.75130 −2.52567 3.35068 −6.00859 1.00000 4.42319
1.5 −2.51571 1.00000 4.32879 −2.36751 −2.51571 −4.50590 −5.85857 1.00000 5.95597
1.6 −2.43983 1.00000 3.95278 1.74573 −2.43983 −3.42450 −4.76447 1.00000 −4.25930
1.7 −1.90134 1.00000 1.61508 −3.96068 −1.90134 −1.76877 0.731855 1.00000 7.53059
1.8 −1.72731 1.00000 0.983594 2.35153 −1.72731 −2.41996 1.75565 1.00000 −4.06181
1.9 −1.35602 1.00000 −0.161218 −0.183673 −1.35602 3.62200 2.93065 1.00000 0.249063
1.10 −1.28568 1.00000 −0.347029 −4.43274 −1.28568 0.824336 3.01753 1.00000 5.69909
1.11 −1.26831 1.00000 −0.391391 1.73153 −1.26831 0.989351 3.03302 1.00000 −2.19612
1.12 −0.697261 1.00000 −1.51383 −2.38011 −0.697261 −0.404514 2.45006 1.00000 1.65956
1.13 −0.308133 1.00000 −1.90505 −0.0777101 −0.308133 0.667754 1.20328 1.00000 0.0239450
1.14 −0.266702 1.00000 −1.92887 3.43095 −0.266702 −1.86360 1.04784 1.00000 −0.915042
1.15 −0.198248 1.00000 −1.96070 −1.60317 −0.198248 −4.39668 0.785201 1.00000 0.317825
1.16 0.466852 1.00000 −1.78205 2.19007 0.466852 1.56308 −1.76566 1.00000 1.02244
1.17 0.783477 1.00000 −1.38616 −2.97805 0.783477 3.18728 −2.65298 1.00000 −2.33323
1.18 0.796821 1.00000 −1.36508 −3.75576 0.796821 −2.20249 −2.68136 1.00000 −2.99267
1.19 1.13447 1.00000 −0.712982 0.951442 1.13447 −0.520635 −3.07779 1.00000 1.07938
1.20 1.22201 1.00000 −0.506690 −3.06594 1.22201 −1.45560 −3.06320 1.00000 −3.74661
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8619.2.a.bs ✓ 24
13.b even 2 1 8619.2.a.bx yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8619.2.a.bs ✓ 24 1.a even 1 1 trivial
8619.2.a.bx yes 24 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{24} + 11 T_{2}^{23} + 22 T_{2}^{22} - 178 T_{2}^{21} - 820 T_{2}^{20} + 449 T_{2}^{19} + \cdots - 251 \) Copy content Toggle raw display
\( T_{5}^{24} + 23 T_{5}^{23} + 189 T_{5}^{22} + 394 T_{5}^{21} - 3501 T_{5}^{20} - 22876 T_{5}^{19} + \cdots + 770861 \) Copy content Toggle raw display
\( T_{7}^{24} + 10 T_{7}^{23} - 35 T_{7}^{22} - 666 T_{7}^{21} - 617 T_{7}^{20} + 16562 T_{7}^{19} + \cdots - 524581 \) Copy content Toggle raw display